They look like asymptotes, don't they. However, if you use Exclusions and ExclusionsStyle, you can get the real asymptotes to show up (in different colors or dashing, if you want).
Back in the dark ages, when you had to write your own graphics code, the appearance of the nearly vertical lines when plotting Tan or Cot was evidence of a bug or sloppy coding. What you do (or did) when plotting a function was to set a sampling of points, evaluate the function at the points, and draw line segments between them. It was easy to miss the fact that the line went from a vary large positive ordinate to a very large negative ordinate, and suppress the line if this was the case. The adaptive sampling in Wolfram Language is very good, but it, by itself, does not look for singularities. This was evidently the case for Mathematica 1 through 4. The bug was fixed in Mathematica 5, and then Mathematica 6 switched from Postscript-based graphics to a more powerful system.
Besides, what you want when you ask Mathematica (Wolfram Language) to plot a function is to see a locus of points for that function, not any asymptotes, etc. The vertical lines are NOT* part of the Tangent function, and should not be drawn unless you also ask for the asymptotes.*
For some functions, Mathematica will make reasonable assumptions about exclusion points -- something it does much better in Plot3D. What I have repeated asked the Wolfram programmers to do is to make sure that the code already in place to detect singularities works in the case of the elementary trig functions. The language already handles the more complicated analogues for Plot3D and many other functions with singularities in the 2D plots.
So, to answer your question: they look like asymptotes, but we did not ask for the plot of the function plus asymptotes. Wolfram language already has an option to draw asymptotes, should we want them.